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Hill’s Equation with Random Forcing Parameters: Determination of Growth Rates Through Random Matrices

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Abstract

This paper derives expressions for the growth rates for the random 2×2 matrices that result from solutions to the random Hill’s equation. The parameters that appear in Hill’s equation include the forcing strength q k and oscillation frequency λ k . The development of the solutions to this periodic differential equation can be described by a discrete map, where the matrix elements are given by the principal solutions for each cycle. Variations in the (q k ,λ k ) lead to matrix elements that vary from cycle to cycle. This paper presents an analysis of the growth rates including cases where all of the cycles are highly unstable, where some cycles are near the stability border, and where the map would be stable in the absence of fluctuations. For all of these regimes, we provide expressions for the growth rates of the matrices that describe the solutions.

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References

  1. Adams, F.C., Bloch, A.M.: Hill’s equation with random forcing terms. SIAM J. Appl. Math. 68, 947–980 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, F.C., Bloch, A.M.: Hill’s equation with random forcing parameters: The limit of delta function barriers. J. Math. Phys. 50, 073501 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  3. Adams, F.C., Bloch, A.M., Butler, S.C., Druce, J.M., Ketchum, J.A.: Orbits and instabilities in a triaxial cusp potential. Astrophys. J. 670, 1027–1047 (2007)

    Article  ADS  Google Scholar 

  4. Cambronero, S., Rider, B., Ramírez, J.: On the shape of the ground state eigenvalue density of a random Hill’s equation. Commun. Pure Appl. Math. 59, 935–976 (2006)

    Article  MATH  Google Scholar 

  5. Carmona, R., Lacroix, J.: Spectral Theory of Random Schroedinger Operators. Birkhauser, Boston (1990)

    MATH  Google Scholar 

  6. Cohen, J.E., Newman, C.M.: The stability of large random matrices and their products. Ann. Probab. 12, 283–310 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Furstenberg, H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  8. Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hill, G.W.: On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon. Acta. Math. 8, 1–36 (1886)

    Article  MathSciNet  Google Scholar 

  10. Lifshitz, I., Gredeskul, S., Pastur, L.: Introduction in the Theory of Disordered Systems. Wiley, New York (1988)

    Google Scholar 

  11. Lima, R., Rahibe, M.: Exact Lyapunov exponent for infinite products of random matrices. J. Phys. A, Math. Gen 27, 3427–3437 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Magnus, W., Winkler, S.: Hill’s Equation. Wiley, New York (1966)

    MATH  Google Scholar 

  13. Oseledec, V.I.: A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)

    MathSciNet  Google Scholar 

  14. Pastur, L., Figotin, A.: Spectra of Random and Almost-Periodic Operators. Series of Comprehensive Studies in Mathematics. Springer, Berlin (1991)

    Google Scholar 

  15. Pincus, S.: Strong laws of large numbers for products of random matrices. Trans. Am. Math. Soc. 287, 65–89 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Michigan Center for Theoretical Physics, Physics Department, University of Michigan, Ann Arbor, MI, 48109, USA

    Fred C. Adams & Anthony M. Bloch

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  1. Fred C. Adams
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  2. Anthony M. Bloch
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Correspondence to Anthony M. Bloch.

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Adams, F.C., Bloch, A.M. Hill’s Equation with Random Forcing Parameters: Determination of Growth Rates Through Random Matrices. J Stat Phys 139, 139–158 (2010). https://doi.org/10.1007/s10955-010-9931-x

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  • DOI: https://doi.org/10.1007/s10955-010-9931-x

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